98 Logic as a Tool natural numbers, division in the domain of reals, or the function “the daughter of _” in the domain of humans This problem has an easy (albeit artificial) fix, good enough for our formal purposes: we designate an element u (for undefined) from the domain and make the function total by assigning u to be its value for all (tuples of) arguments for which it is not defined For instance, we can extend division by in the domain of reals by putting x/0 = 17 for any real x This may sound reckless, but if proper care is taken when reasoning about division, it will not lead to confusion or contradiction Constants Some objects in the domain can be distinguished in a way that would allow us to make direct references to them, by giving them names Such distinguished √ objects are called constants2 Examples in the domain of real numbers are 0, 1, 3/7, 2, π, e, etc Note that the notion of a constant pertains to the language rather than to the domain For instance, in common calculus we not have a special name for the least positive solution of the equation cos x = x, but if for some purposes we need to refer directly to it then we can extend our mathematical language by giving it a name, that is, make it a constant in our domain We are now ready to define the general notion of a first-order structure Definition 77 A first-order structure (hereafter, just structure) consists of a non-empty domain and a family of distinguished functions, predicates, and constants in that domain Example 78 Here are some examples of structures that will be used further • N , with domain being the set of natural numbers N, the unary function s (successor, i.e., s(x) = x + 1), the binary functions + (addition) and × (multiplication), the predicates =, , and the constant • Z: as for N , but with the domain being the set of integers Z and the additional function − (subtraction) • With the same functions and predicates we take the domain to be the set of rational numbers Q or the reals R The resulting structures are denoted Q and R, respectively (For those familiar with some abstract algebra: algebraic structures such as groups, rings, and fields are all examples of first-order structures Algebraic structures usually only involve functions and constants but no predicates, except = and possibly